Abstract
In this paper we discuss two issues related to model reduction of deterministic or stochastic processes. The first is the relationship of the spectral properties of the dynamics on the attractor of the original, high-dimensional dynamical system with the properties and possibilities for model reduction. We review some elements of the spectral theory of dynamical systems. We apply this theory to obtain a decomposition of the process that utilizes spectral properties of the linear Koopman operator associated with the asymptotic dynamics on the attractor. This allows us to extract the almost periodic part of the evolving process. The remainder of the process has continuous spectrum. The second topic we discuss is that of model validation, where the original, possibly high-dimensional dynamics and the dynamics of the reduced model – that can be deterministic or stochastic – are compared in some norm. Using the “statistical Takens theorem” proven in (Mezić, I. and Banaszuk, A. Physica D, 2004) we argue that comparison of average energy contained in the finite-dimensional projection is one in the hierarchy of functionals of the field that need to be checked in order to assess the accuracy of the projection.
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References
Adrover, A. and Giona, M., ‘Modal reduction of PDE models by means of snapshot archetypes’, Physica D 182, 2003, 23–45.
Akhiezer, N. I. and Glazman, I. M., Theory of Linear Operators in Hilbert Space, Vol. 1. Frederic Ungar Publishing Company, New York, 1961.
Arnold, L. and Crauel, H., ‘Random dynamical systems’, in Springer-Verlag Lecture Notes in Mathematics, Vol. 1486, 1991, pp. 1–22.
Bamieh, B. and Dahleh, M., ‘Energy amplification in channel flows with channel excitation’, Physics of Fluids 13, 2001, 3258–3269.
Berkooz, G., ‘An observation on probability density equations, or, when do simulations reproduce statistics?’, Nonlinearity 7, 1994, 313–328.
Broer, H. and Takens, F., ‘Mixed spectra and rotational symmetry’, Archives of Rational Mechanics Analysis 124, 1993, 13–42.
Butler, K. M. and Farrell, B. F., ‘Three-dimensional optimal perturbations in viscous shear flow’, Physics of Fluids A 4, 1992, 1637.
Chorin, A. J., Hald, O. H., and Kupferman, R., ‘Optimal prediction and the Mori–Zwanzig representation of irreversible processes’, in Proceedings of the National Academy of Sciences of the United States of America, Vol. 97, 2000, pp. 2968–2973.
Dellnitz, M., Froyland, G., and Sertl, S., ‘On the isolated spectrum of the Perron–Frobenius operator’, Nonlinearity 13, 2000, 1171–1188.
Dellnitz, M. and Junge, O., ‘On the approximation of complicated dynamical behavior’, SIAM Journal on Numerical Analysis 36(2), 1999, 491–515.
Farge, M. and Schneider, K., ‘Coherent vortex simulation (cvs), a semi-deterministic turbulence model using wavelets’, Flow Turbulence and Combustion 66, 2001, 393–426.
Givon, D., Kupferman, R., and Stuart, A., Extracting Macroscopic Dynamics: Model Problems and Algorithms. Warwick University, Preprint, 2003.
Gustavsson, L. H., ‘Energy growth of three-dimensional disturbances in plane Poiseuille flow’, Journal of Fluid Mechanics 224, 1991, 241–260.
Holmes, P., Lumley, J. L., and Berkooz, G., Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge University Press, Cambridge, MA, 1996.
Majda, A. J., Timofeyev, I., and Eijnden, E. V., ‘Models for stochastic climate prediction’, in Proceedings of the National Academy of Sciences, Vol. 96, 1999, pp. 14687–14691.
Mane, R., Ergodic Theory and Differentiable Dynamics, Springer-Verlag, 1987.
Mezić, I. I. and Banaszuk, A., Comparison of complex systems. Physica D, 2004.
Mezić, I. I. and Wiggins, S., ‘A method for visualization of invariant sets of dynamical systems based on the ergodic partition’, Chaos 9, 1999, 213–218.
Petersen, K., Ergodic Theory, Cambridge University Press, Cambridge, MA, 1995.
Plesner, A. I., Spectral Theory of Linear Operators, Vol. 2, Frederic Ungar Publishing Company, New York, 1969.
Reddy, S. C. and Henningson, D. S., ‘Energy growth in viscous channel flows’, Journal of Fluid Mechanics 252, 1993, 209.
Reynolds, W. C. and Hussein, A. K. M. F., ‘The mechanics of an organized wave in turbulent shear flow: Part 3. Theoretical models and comparison with experiment’, Journal of Fluid Mechanics 54, 1972, 263–288.
Rokhlin, V. A., ‘Selected topics from the metric theory of dynamical systems’, American Mathematical Society Translation, Series 2 49, 1966, 171–240.
Rowley, C. W., Kevrekidis, I. G., Marsden, J. E., and Lust, K., ‘Reduction and reconstruction for self-similar dynamical systems’, Nonlinearity 16, 2003, 1257–1275.
Saravanan, R. and McWilliams, J. C., ‘Stochasticity and spatial resonance in interdecadal climate fluctuations’, Journal of Climate 10, 1997, 2299–2320.
Sinai, Ya. G., Topics in Ergodic Theory, Princeton Univerity Press, Princeton, NJ, 1994.
Temam, R., Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997.
Trefethen, L. N., Trefethen, A. E., Reddy, S. C., and Driscoll, T. A., ‘Hydrodynamic stability without eigenvalues’, Science 261, 1993, 578.
Wiener, N. and Wintner, A., ‘Harmonic analysis and ergodic theory’, American Journal of Mathematics 63, 1940.
Yosida, K., Functional Analysis, Springer-Verlag, New York, 1980.
Young, L. S., ‘What are SRB measures, and which dynamical systems have them?’, Journal of Statistical Physics 108, 2002, 733–754.
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Mezić, I. Spectral Properties of Dynamical Systems, Model Reduction and Decompositions. Nonlinear Dyn 41, 309–325 (2005). https://doi.org/10.1007/s11071-005-2824-x
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DOI: https://doi.org/10.1007/s11071-005-2824-x